Probably the best single answer is Poincare. Vanden Eynde's history ofhomotopy introduces the universal cover as a part of Poincare's 1883work on uniformization, using analytic continuation. ("Development ofthe concept of homotopy" in I, M, James HISTORY OF TOPOLOGY, p. 82).That would involve both aspects that you asked about.

Vanden Eynde (so far as I can see) does not really say Poincare was thefirst. Probably it is just too complicated a question when you go intodetail.

The themes go back to Abel and Cauchy on Abelian integrals and analyticcontinuation--in hindsight that was all about connecting paths ofintegration to series (with radii of convergence) and so to patchescovering a domain. It was not clearly understood in terms of coveringsurfaces until Riemann, and then people took decades to get clear onRiemann surfaces.